WASHINGTON — The former acting F.B.I. director, Andrew G. McCabe, wrote a confidential memo last spring recounting a conversation that offered significant behind-the-scenes details on the firing of Mr. McCabe’s predecessor, James B. Comey, according to several people familiar with the discussion.

Mr. Comey’s firing is a central focus of the special counsel’s investigation into whether President Trump tried to obstruct the investigation into his campaign’s ties to Russia. Mr. McCabe has turned over his memo to the special counsel, Robert S. Mueller III.

In the document, whose contents have not been previously reported, Mr. McCabe described a conversation at the Justice Department with the deputy attorney general, Rod J. Rosenstein, in the chaotic days last May after Mr. Comey’s abrupt firing. Mr. Rosenstein played a key role in the dismissal, writing a memo that rebuked Mr. Comey over his handling of an investigation into Hillary Clinton.

But in the meeting at the Justice Department, Mr. Rosenstein added a new detail: He said the president had originally asked him to reference Russia in his memo, the people familiar with the conversation said. Mr. Rosenstein did not elaborate on what Mr. Trump had wanted him to say.

To Mr. McCabe, that seemed like possible evidence that Mr. Comey’s firing was actually related to the F.B.I.’s investigation into the Trump campaign’s ties to Russia, and that Mr. Rosenstein helped provide a cover story by writing about the Clinton investigation.

One person who was briefed on Mr. Rosenstein’s conversation with the president said Mr. Trump had simply wanted Mr. Rosenstein to mention that he was not personally under investigation in the Russia inquiry. Mr. Rosenstein said it was unnecessary and did not include such a reference. Mr. Trump ultimately said it himself when announcing the firing.

Mr. McCabe’s memo, one of several that he wrote, highlights the conflicting roles that Mr. Rosenstein plays in the case. He supervises the special counsel investigation and has told colleagues that protecting it is among his highest priorities. But many current and former law enforcement officials are suspicious of some of his other actions, including allowing some of Mr. Trump’s congressional allies to view crucial documents from the investigation.

In conversations with prosecutors, Mr. Trump’s lawyers have cited Mr. Rosenstein’s involvement in the firing of Mr. Comey as proof that it was not an effort to obstruct justice, according to people familiar with the president’s legal strategy.

That argument has only made Mr. Rosenstein’s position even more peculiar: He oversees an investigation into the president, who points to Mr. Rosenstein’s own actions as evidence that he is innocent. And Mr. Rosenstein could have the final say on whether that argument has merit.

The people who discussed the meeting and the memo did so on the condition of anonymity because they were not authorized to publicly discuss the matters. A spokeswoman for Mr. McCabe declined to comment. Mr. McCabe was fired in March after a finding that he was not candid in an internal investigation. Mr. McCabe has said the firing was a politically motivated effort to discredit him as a witness in the special counsel investigation.

A Justice Department spokeswoman also declined to comment. Mr. Rosenstein has consulted departmental ethics advisers about whether to recuse himself from the Russia investigation and has not done so.

“I’ve talked with Director Mueller about this,” Mr. Rosenstein told The Associated Press last year. “He’s going to make the appropriate decisions, and if anything that I did winds up being relevant to his investigation then, as Director Mueller and I discussed, if there’s a need from me to recuse, I will.”

Removing Mr. Rosenstein from the investigation, though, would only add uncertainty to the process. He is regarded, even among his critics, as a bulwark against an effort by Mr. Trump to fire Mr. Mueller and shut down the investigation. Mr. Trump has openly mused about doing so, and has considered firing Mr. Rosenstein, too.

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Memos written by Andrew G. McCabe, the former acting F.B.I. director, highlight the conflicting roles that Mr. Rosenstein plays in the special counsel investigation.CreditJim Watson/Agence France-Presse — Getty Images

Mr. McCabe’s memo reflects the F.B.I.’s early efforts to discern Mr. Trump’s intentions in firing Mr. Comey, an effort that continues today. Mr. Trump and his advisers have issued conflicting and changing explanations for the termination.

At first, they pointed to Mr. Rosenstein’s reasoning, which criticized Mr. Comey’s handling of the Clinton investigation. He was unusually public about the inquiry in ways that Democrats say contributed to Mrs. Clinton’s defeat.

But Mr. Trump quickly undercut that statement, telling NBC News that he had planned to fire Mr. Comey even before receiving Mr. Rosenstein’s memo. “And in fact, when I decided to just do it, I said to myself, I said, ‘You know, this Russia thing with Trump and Russia is a made-up story,’” Mr. Trump said. “It’s an excuse by the Democrats for having lost an election that they should have won.”

Mr. Rosenstein’s comments to Mr. McCabe were made against a backdrop of those shifting explanations. After their meeting, Mr. Rosenstein gave Mr. McCabe a copy of a draft firing letter that Mr. Trump had written, according to two people familiar with the conversation. Mr. McCabe later gave that letter, and his memos, to Mr. Mueller.

Mr. McCabe’s memo reflects the anxiety of the early months of the Trump administration and presaged a relationship with law enforcement that has only grown more strained. Just as Mr. Comey kept memos on interactions with Mr. Trump and Attorney General Jeff Sessions, Mr. McCabe documented his own conversations with the president and others.

Mr. Trump has injected himself into Justice Department operations in ways that have little precedent. While most presidents who have faced federal investigations have assiduously avoided discussing them for fear of being seen as trying to influence them, Mr. Trump has shown no hesitation. He has called the investigation a “witch hunt,” declared that a “deep state” was trying to undermine his presidency, and encouraged the Justice Department to provide sensitive details about the special counsel inquiry to Congress.

Most recently, Mr. Trump has publicly demanded that the Justice Department investigate the Russia investigation itself.

In response, Mr. Rosenstein has walked a perilous line. Faced with threats on his job, he told Republicans in Congress that he would not be “extorted.” But he has also relented to pressure in some instances, providing information to Congress that would not normally be shared amid an investigation.

And in response to the president’s calls for an investigation into whether the F.B.I. used informants to infiltrate his campaign — a charge for which there is no public evidence — Mr. Rosenstein referred the matter to the inspector general and issued a public statement that some current and former officials said was too tepid.

“If anyone did infiltrate or surveil participants in a presidential campaign for inappropriate purposes, we need to know about it and take appropriate action,” Mr. Rosenstein said.

Mr. Rosenstein has said little about his strategy for dealing with the political crosswinds. But he has defended his memo about Mr. Comey. “I wrote it. I believe it. I stand by it,” he said in a statement last year. He added that it was never intended to “justify a for-cause termination.”

Recently, Rudolph W. Giuliani, the president’s lawyer, added a new explanation for Mr. Comey’s firing. He said Mr. Trump was upset that Mr. Comey would not publicly clear him in the Russia investigation.

“He fired Comey because Comey would not, among other things, say that he wasn’t a target of the investigation,” Mr. Giuliani said.

A version of this article appears in print on , on Page A16 of the New York edition with the headline: In Memo, F.B.I. Official Suspects a Cover Story. Order Reprints | Today’s Paper | Subscribe